(Certain) colimit and product in category of topological spaces

Question

Consider the diagram

$$(*):\;\;\;X_0 \stackrel{i_0}\hookrightarrow X_1 \stackrel{i_1}\hookrightarrow X_2 \stackrel{i_2}\hookrightarrow \cdots $$

in category of topological spaces. Denote $I$ the unit interval $[0,1]$ with the standard topology.

Then we also have

$$(*)\times I:\;\;\;X_0\times I \stackrel{i_0\times 1_I}\hookrightarrow X_1 \times I \stackrel{i_1\times 1_I}\hookrightarrow X_2 \times I \hookrightarrow \cdots $$

and colimits of those two diagrams, which are basically $$\mathrm{colim}(*)=\left(\coprod_{n =0}^\infty X_n\right)/{\sim}, \;\;\; x \sim i_j(x), \; j \in \mathbb{N}, \; x \in X_j,$$ $$\mathrm{colim}((*)\times I)=\left(\coprod_{n =0}^\infty X_n \times I\right)/{\sim}, \;\;\; (x,t) \sim (i_j(x),t), \; j \in \mathbb{N}, \; x \in X_j, t \in I.$$

(Question 0: is this correct?)

I am interested in the map $$f:(\operatorname{colim}(*))\times I \rightarrow \mathrm{colim}((*)\times I)$$ defined as $([x],t)\mapsto [(x,t)]$, where $[-]$ denotes taking the equivalence classes with respect to the considered equivalences. Obviously it is correctly defined as a map.

Question 1: Is this map continous? That is, are the two objects homeomorphic? (I believe one can obtain the inverse map $f^{-1}$ using the universal property of $\mathrm{colim}((*)\times I)$, hence $f^{-1}$ is continuous).

I have more information about the maps $i_j$. For example, they are closed maps.

This is a part of a homework assignment, so I would appreciate hints instead of full answer.

(Maybe some context and motivation: My goal is to define a homotopy $F:\mathrm{colim}(*))\times I \rightarrow \mathrm{colim}(*)$ using easy-to-describe homotopies $F_n:X_n\times I \rightarrow X_{n+1}$, but these induce "only" a map $(\operatorname{colim }F_n): \operatorname{colim}((*)\times I) \rightarrow \operatorname{colim}(*)$, so I am trying to change the domain somehow - so ultimately, I need only the composite map $(\operatorname{colim }F_n) \circ f$ to be continuous.)

Answer 1

I asked a very similar question some time ago. The key fact is the following:

Let $X,Y$ be a topological spaces. Suppose $X$ comes with an equivalence relation $\mathcal{R}\subset X\times X$. Then the canonical map $$\frac{X\times Y}{\mathcal R\times \mathsf{1}}\longrightarrow \frac{X}{\mathcal{R}}\times Y,$$ is a continuous bijection (that may fail to be a homeomorphism). However, if $Y$ is locally compact Hausdorff, then it is a homeomorphism.

Here $\mathcal R\times \mathsf{1}$ designates the equivalence relation on $X\times Y$ whereby $(x,y)\sim (x',y')$ iff $x\mathcal{R}x'$ and $y=y'$. The proof of continuity is easy, it follows immediately form the universal properties of quotient spaces and products. To finish off your question, let $X=\coprod_nX_n$, $Y=I$ and $\mathcal R$ the equivalence relation defining the colimit of your first diagram. Then the colimit of your second diagram is the quotient space of $X\times I$ (because $\left(\coprod_nX_n\right)\times I$ is canonically isomorphic to $\coprod_n\left(X_n\times I\right)$) by the equivalence relation $\mathcal R\times\mathsf{1}$. The facts from the highlighted paragraph then tell us that the comparison map you defined is a homeomorphism.

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This can be extended. The key fact is that the functor $$-\times I:\mathbf{Top}\to\mathbf{Top}$$ has a right adjoint $$\mathbf{Hom}(I,-):\mathbf{Top}\to\mathbf{Top}$$ where $\mathbf{Hom}(I,-)$ sends a space $X$ to the set of all continuous maps $I\to X$ equipped with the compact open topology. The unit interval is core compact (being locally compact Hausdorff). Having a right adjoint, the functor $-\times I$ commutes to all colimits, that is there are isomorphisms in Top (in other words, a homeomorphism) $$\mathrm{colim}_{\,\mathcal{I}}(F)\times I\to\mathrm{colim}_{\,\mathcal{I}}((-\times I)\circ F)$$ for any small diagram $F:\mathcal I\to\mathbf{Top}$.

Answer 2

There is a natural continuous map $\varphi:\text{colim}(X_n×I)\to\text{colim}(X_n)\times I$ whose existence is indicated by the following diagram

$$\begin{array}{ccc} \coprod(X_n×I) & → & \text{colim}(X_n×I) \\ \downarrow & & \downarrow\\ \left(\coprod X_n\right)×I & \xrightarrow{q×1} & \text{colim}(X_n)×I \end{array}$$ where the left vertical map is induced by $X_n×I→\left(\coprod X_n\right)×I$. It's a standard exercise to show that the left map is a homeomorphism, so it doesn't really matter if we regard some $(x,t)$ an element of the first or the second space. Then $φ$ is induced as a continuous map from the quotient because if two points $(x,t)$ and $(y,s)$ are identified via the generating relations, then $s=t$ and $y=i_j(x)$, thus $[x]=[y]$. This $φ$ sends $[(x,t)]$ to $([x],t)$ and it is easy to show that $φ$ is bijective. This gives us the inverse $\psi$ which send $([x],t)$ to $[(x,t)]$, but there doesn't seem to be anything which suggests continuity of $ψ$.
The trick is to show that the map $q×1$ is a quotient map, as this will make the continuous bijection $φ$ a homeomorphism. There are two proofs for this fact, one more direct and elementary proof, and another proof which is more categorical and involves more diagrams and less topology, but both use the fact that $I$ is locally compact at some point.

Answer 3

There is a very nice proof in terms of adjoint functors. We want to test the conjectured isomorphism against all topological spaces, that is $$ \textrm{Hom}_{Top} \left ( \varinjlim_{d \in D } X(d) \times I , Y \right ) \stackrel{?}{\simeq} \textrm{Hom}_{Top} \left ( \left (\varinjlim_{d \in D } X(n) \right ) \times I , Y \right )$$ Here $D$ is whatever category, $X: D \to \textrm{Top}$ is whatever diagram of spaces, and $Y$ is any space.

Our strategy is to use adjoint functors and bring everything on the right side of the hom, then check if we actually have an isomorphism. This could seem very clever the first time, but it's sort of the unique passage you can do to do some manipulations.

First thing we have to be able to "bring on the other side" colimits: this comes from the adjunction $$ \textrm{Hom}_{Top} \left ( \varinjlim_{d \in D } F(d) , Z \right ) \simeq \textrm{Nat}_{D} ( F, \Delta(Z))$$ for any functor $F: D \to \textrm{Top}$ and space $Z$, where $\Delta: \textrm{Top} \to \textrm{Top}^{\mathbb{N}} $ is the constant functor.

Second, we have to be able to bring on the other side the $_ \times I$ operation. This is possible since $I$ is locally compact and the exponential law holds: $$ \textrm{Hom}_{\textrm{Top}}(A \times I, B) \simeq \textrm{Hom}_{\textrm{Top}}(A , B^I) $$ The general theorem is not easy, but in this special case (that is, the space being the interval) you should be able to prove it yourself.

Now let's put everything together: we have $$\textrm{Hom}_{Top} \left ( \varinjlim_{d \in D } X(d) \times I , Y \right ) \simeq \textrm{Nat}_{D} ( X \times I , \Delta(Y)) \simeq \textrm{Nat}_{D} ( X , \Delta(Y)^I) $$ Now note that taking maps from $I$ to $\Delta(Y)$ componentwise is the same of taking the constant functor $\Delta(Y^{I})$, since $\Delta(Y)$ is a constant functor!! Thus we can continue, using adjunctions in the other way around: $$ \simeq \textrm{Nat}_{D} ( X , \Delta(Y^I)) \simeq \textrm{Hom}_{Top} \left ( \varinjlim_{d \in D } X(d) , Y^I \right ) \simeq \textrm{Hom}_{Top} \left ( \left ( \varinjlim_{d \in D } X(d) \right ) \times I , Y \right ) $$

Wow, done! As you can see, the (hard) commutation of colim and $\times I$ becomes the (easy) commutation of $\Delta$ and $(-)^I$ in the adjoint point of view.